286 research outputs found
Extended Integrated Interleaved Codes over any Field with Applications to Locally Recoverable Codes
Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes
are a versatile alternative for Locally Recoverable (LRC) codes, since they
require fields of relatively small size. II and EII codes are generally defined
over Reed-Solomon type of codes. A new comprehensive definition of EII codes is
presented, allowing for EII codes over any field, and in particular, over the
binary field . The traditional definition of II and EII codes is shown
to be a special case of the new definition. Improvements over previous
constructions of LRC codes, in particular, for binary codes, are given, as well
as cases meeting an upper bound on the minimum distance. Properties of the
codes are presented as well, in particular, an iterative decoding algorithm on
rows and columns generalizing the iterative decoding algorithm of product
codes. Two applications are also discussed: one is finding a systematic
encoding of EII codes such that the parity symbols have a balanced distribution
on rows, and the other is the problem of ordering the symbols of an EII code
such that the maximum length of a correctable burst is achieved.Comment: 25 page
Integrated Interleaved Codes as Locally Recoverable Codes: Properties and Performance
Considerable interest has been paid in recent literature to codes combining
local and global properties for erasure correction. Applications are in cloud
type of implementations, in which fast recovery of a failed storage device is
important, but additional protection is required in order to avoid data loss,
and in RAID type of architectures, in which total device failures coexist with
silent failures at the page or sector level in each device. Existing solutions
to these problems require in general relatively large finite fields. The
techniques of Integrated Interleaved Codes (which are closely related to
Generalized Concatenated Codes) are proposed to reduce significantly the size
of the finite field, and it is shown that when the parameters of these codes
are judiciously chosen, their performance may be competitive with the one of
codes optimizing the minimum distance.Comment: 24 pages, 5 figures and 3 table
Extended Product and Integrated Interleaved Codes
A new class of codes, Extended Product (EPC) Codes, consisting of a product
code with a number of extra parities added, is presented and applications for
erasure decoding are discussed. An upper bound on the minimum distance of EPC
codes is given, as well as constructions meeting the bound for some relevant
cases. A special case of EPC codes, Extended Integrated Interleaved (EII)
codes, which naturally unify Integrated Interleaved (II) codes and product
codes, is defined and studied in detail. It is shown that EII codes often
improve the minimum distance of II codes with the same rate, and they enhance
the decoding algorithm by allowing decoding on columns as well as on rows. It
is also shown that EII codes allow for encoding II codes with an uniform
distribution of the parity symbols.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1610.0427
Multi-Erasure Locally Recoverable Codes Over Small Fields
Erasure codes play an important role in storage systems to prevent data loss.
In this work, we study a class of erasure codes called Multi-Erasure Locally
Recoverable Codes (ME-LRCs) for storage arrays. Compared to previous related
works, we focus on the construction of ME-LRCs over small fields. We first
develop upper and lower bounds on the minimum distance of ME-LRCs. Our main
contribution is to propose a general construction of ME-LRCs based on
generalized tensor product codes, and study their erasure-correcting
properties. A decoding algorithm tailored for erasure recovery is given, and
correctable erasure patterns are identified. We then prove that our
construction yields optimal ME-LRCs with a wide range of code parameters, and
present some explicit ME-LRCs over small fields. Finally, we show that
generalized integrated interleaving (GII) codes can be treated as a subclass of
generalized tensor product codes, thus defining the exact relation between
these codes.Comment: This is an extended version of arXiv:1701.06110. To appear in
Allerton 201
On Locally Recoverable (LRC) Codes
We present simple constructions of optimal erasure-correcting LRC codes by
exhibiting their parity-check matrices. When the number of local parities in a
parity group plus the number of global parities is smaller than the size of the
parity group, the constructed codes are optimal with a field of size at least
the length of the code. We can reduce the size of the field to at least the
size of the parity groups when the number of global parities equals the number
of local parities in a parity group plus one.Comment: 11 pages, 2 figure
Multiple-Layer Integrated Interleaved Codes: A Class of Hierarchical Locally Recoverable Codes
The traditional definition of Integrated Interleaved (II) codes generally
assumes that the component nested codes are either Reed-Solomon (RS) or
shortened Reed-Solomon codes. By taking general classes of codes, we present a
recursive construction of Extended Integrated Interleaved (EII) codes into
multiple layers, a problem that brought attention in literature for II codes.
The multiple layer approach allows for a hierarchical scheme where each layer
of the code provides for a different locality. In particular, we present the
erasure-correcting capability of the new codes and we show that they are
ideally suited as Locally Recoverable (LRC) codes due to their hierarchical
locality and the small finite field required by the construction. Properties of
the multiple layer EII codes, like their minimum distance and dimension, as
well as their erasure decoding algorithms, parity-check matrices and
performance analysis, are provided and illustrated with examples. Finally, we
will observe that the parity-check matrices of high layer EII codes have low
density.Comment: 21 pages, 1 tabl
Hierarchical Hybrid Error Correction for Time-Sensitive Devices at the Edge
Computational storage, known as a solution to significantly reduce the
latency by moving data-processing down to the data storage, has received wide
attention because of its potential to accelerate data-driven devices at the
edge. To meet the insatiable appetite for complicated functionalities tailored
for intelligent devices such as autonomous vehicles, properties including
heterogeneity, scalability, and flexibility are becoming increasingly
important. Based on our prior work on hierarchical erasure coding that enables
scalability and flexibility in cloud storage, we develop an efficient decoding
algorithm that corrects a mixture of errors and erasures simultaneously. We
first extract the basic component code, the so-called extended Cauchy (EC)
codes, of the proposed coding solution. We prove that the class of EC codes is
strictly larger than that of relevant codes with known explicit decoding
algorithms. Motivated by this finding, we then develop an efficient decoding
method for the general class of EC codes, based on which we propose the local
and global decoding algorithms for the hierarchical codes. Our proposed hybrid
error correction not only enables the usage of hierarchical codes in
computational storage at the edge, but also applies to any Cauchy-like codes
and allows potentially wider applications of the EC codes.Comment: 29 pages (single column), 0 figures, to be submitted to IEEE
Transactions on Communication
Hierarchical Coding for Cloud Storage: Topology-Adaptivity, Scalability, and Flexibility
In order to accommodate the ever-growing data from various, possibly
independent, sources and the dynamic nature of data usage rates in practical
applications, modern cloud data storage systems are required to be scalable,
flexible, and heterogeneous. The recent rise of the blockchain technology is
also moving various information systems towards decentralization to achieve
high privacy at low costs. While codes with hierarchical locality have been
intensively studied in the context of centralized cloud storage due to their
effectiveness in reducing the average reading time, those for decentralized
storage networks (DSNs) have not yet been discussed. In this paper, we propose
a joint coding scheme where each node receives extra protection through the
cooperation with nodes in its neighborhood in a heterogeneous DSN with any
given topology. This work extends and subsumes our prior work on coding for
centralized cloud storage. In particular, our proposed construction not only
preserves desirable properties such as scalability and flexibility, which are
critical in dynamic networks, but also adapts to arbitrary topologies, a
property that is essential in DSNs but has been overlooked in existing works.Comment: 25 pages (single column), 19 figures, submitted to the IEEE
Transactions on Information Theory (TIT
Generalized Concatenated Types of Codes for Erasure Correction
Generalized Concatenated (GC), also known as Integrated Interleaved (II)
Codes, are studied from an erasure correction point of view making them useful
for Redundant Arrays of Independent Disks (RAID) types of architectures
combining global and local properties. The fundamental erasure-correcting
properties of the codes are proven and efficient encoding and decoding
algorithms are provided. Although less powerful than the recently developed
PMDS codes, this implementation has the advantage of allowing generalization to
any range of parameters while the size of the field is much smaller than the
one required for PMDS codes
Universal and Dynamic Locally Repairable Codes with Maximal Recoverability via Sum-Rank Codes
Locally repairable codes (LRCs) are considered with equal or unequal
localities, local distances and local field sizes. An explicit two-layer
architecture with a sum-rank outer code is obtained, having disjoint local
groups and achieving maximal recoverability (MR) for all families of local
linear codes (MDS or not) simultaneously, up to a specified maximum locality . Furthermore, the local linear codes (thus the localities, local distances
and local fields) can be efficiently and dynamically modified without global
recoding or changes in architecture or outer code, while preserving the MR
property, easily adapting to new configurations in storage or new hot and cold
data. In addition, local groups and file components can be added, removed or
updated without global recoding. The construction requires global fields of
size roughly , for local groups and maximum or specified locality
. For equal localities, these global fields are smaller than those of
previous MR-LRCs when (global parities). For unequal localities,
they provide an exponential field size reduction on all previous best known
MR-LRCs. For bounded localities and a large number of local groups, the global
erasure-correction complexity of the given construction is comparable to that
of Tamo-Barg codes or Reed-Solomon codes with local replication, while local
repair is as efficient as for the Cartesian product of the local codes.
Reed-Solomon codes with local replication and Cartesian products are recovered
from the given construction when and , respectively. The given
construction can also be adapted to provide hierarchical MR-LRCs for all types
of hierarchies and parameters. Finally, subextension subcodes and sum-rank
alternant codes are introduced to obtain further exponential field size
reductions, at the expense of lower information rates
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